wen hou and cishen zhang - · pdf fileabstract—a novel frame of parallel-beam...

Abstract—A novel frame of parallel-beam Computerized

Tomography (CT) reconstruction is proposed. The CT

projections are firstly converted to Mojette projections. Then

1D Fourier transform is applied to the modified Mojette

projections, followed by an exact mapping of the gained

Fourier coefficients to the 2D Fourier domain of the scanning

object. Finally the scanning object can be reconstructed from

the partial Fourier coefficients by compressed sensing (CS).

Experimental results show that using this frame, the purpose of

reducing the radiation dosage during CT examinations without

compromising the image quality can be achieved.

Index Terms—Computerized tomography, parallel-beam,

mojette transform, compressed sensing.

I. INTRODUCTION

Computerized Tomography (CT) is a powerful medical

tool which is helpful for the diagnosis of doctors and thus

can benefit patients a lot. In general, the image quality of CT

is proportional to the radiation dosages. But increased

radiation dosages raise the risk of cancer, which becomes an

obstacle for CT development. Hence, it is meaningful and

challenging work to reduce the radiation dosage during CT

examinations without compromising the image quality.

Classical tomography is concerned with the recovery of

the scanning object from a set of projections, which is the

Radon transform (RT) [1] of the object. As Radon transform

is continuous both for the object under scan and the

projections themselves, it has to be sampled to adjust to

practical application, resulting in ill-posedness. Mojette

transform [2] is a both experimentally and computationally

viable discretization of Radon transform. It regularizes the

ill-posedness of inverse Radon transform and allows for an

exact reconstruction in the discrete domain with a finite

number of projections. Despite its advantages, Mojette

transform has two main shortcomings. One is that it is not

compatible with the physical acquisition of CT, and this

problem is addressed in paper [3], in which the authors

design a linear system to calculate the Mojette projections

from a Radon acquisition. The other is that the angle and bin

set is different from practical CT machine, i.e., classical

tomograph acquires are equally distributed over 2π with a

fixed number of bins onto each projection, while Mojette

transform is defined over Farey angles with varying

orientation and number of bins onto each projection. A lot of

research [4] has been done, committed to applying Mojette

Manuscript received October 20, 2012; revised November 24, 2012.

Wen Hou and Cishen Zhang are with the Faculty of Engineering and

Industrial Sciences, Swinburne University of Technology, Vic 3122,

Australia (e-mail: [email protected]; [email protected])

transform to classical tomographic data, which implies the

advantage of Mojette transform on one hand and improves

the potential of Mojette transform in practical application on

the other hand. Paper [4] claims that any set of real, acquired

tomographic data can be rebinned into a compatible Mojette

projection space, without any loss of reconstruction power.

After getting the projections, the next step is the

reconstruction problem. M. Katz [5] presents some

reconstructubility theorems for discrete images and

projections. A lot of work has also been done to recover

object based on Mojette projections, committed to the

improvement of the accuracy. Since we try to reduce the

radiation dosage as well, a newly developed technique

called compressed sensing (CS) is used. CS attempts to

reconstruct signals from significantly fewer samples than

were traditionally thought necessary (e.g., Nyquist sampling

theorem) [6]. CT imagery happens to meet the two key

requirements for successful application of CS: it is

compressible in some transform domain and the scanner

gets encoded samples. Applying CS to CT reconstruction,

the high-accuracy reconstruction with low-dosage radiation

is expected to be obtained [7].

The main contribution of the paper is that a novel

reconstruction frame, i.e. the combination of Mojette

transform and CS, is proposed in the interest of accuracy

and radiation. A detailed description and explanation about

how to modify the Mojette projections for the application of

CS is given. The rest of paper is organized as follows. In

Section II, the Mojette transform and CS is briefly

introduced, and the relation between Mojette transform and

the 2D Fourier coefficients of the scanning object is

formulated, at last the detailed reconstruction algorithm is

described. Section III gives the experimental results to

validate the accuracy, low-dosage radiation and noise

tolerance of the proposed frame. Conclusions are drawn in

Section IV.

II. RECONSTRUCTION FRAME BASED ON MOJETTE

TRANSFORM AND COMPRESSED SENSING

A. Mojette Transform

For simplicity, we assume the scanning object to be

𝑓(𝑥, 𝑦) with the size 𝑁 × 𝑁 (𝑁 is even). CT scanner maps

the 2D object into a set of 1D projection lines, which forms

the sinogram. Each line is the Radon transform of 𝑓 for a

given angle θ and a module ρ [1], defined by:

𝑅𝜃 𝜌 = 𝑓(𝑥, 𝑦)𝛿(𝜌 − 𝑥𝑐𝑜𝑠𝜃 − 𝑦𝑠𝑖𝑛𝜃)+∞

−∞

+∞

−∞𝑑𝑥𝑑𝑦 (1)

Parallel-Beam CT Reconstruction Based on Mojette

Transform and Compressed Sensing

Wen Hou and Cishen Zhang

83DOI: 10.7763/IJCEE.2013.V5.669

International Journal of Computer and Electrical Engineering, Vol. 5, No. 1, February 2013

where θ and ρ are respectively the angular and radial

coordinates of the projection line (𝜃, 𝜌), and 𝛿 is the Dirac's

delta function.

The Dirac-Mojette transform is an exact discretization of

the Radon transform. It is defined over angles =𝑡𝑎𝑛−1(𝑞/𝑝) , where 𝑝 and 𝑞 are relatively prime integers.

As the Farey series of order 𝐾, denoted by 𝐹𝑘 , is the set of

all fractions in lowest terms between 0 and ∞ , whose

denominators don't exceed 𝐾 , e.g.

𝐹4 = [0

1,

1

4,

1

3,

1

2,

2

3,

3

4,

1

1,

4

3,

3

2,

2

1,

3

1,

4

1,

1

0], we can use it to give a

set of discrete angles between [0,𝜋

2] and obtain the angles

over [0, 𝜋] by symmetry. The Dirac-Mojette transform is

defined as:

𝑀𝑝 ,𝑞 𝜌 = 𝑓(𝑥, 𝑦)∆(𝜌 − 𝑝𝑥 − 𝑞𝑦)𝑦𝑥 (2)

The geometry of Dirac-Mojette projector is shown in

Fig.1(a), where the pixel is summed to its corresponding bin

if and only if the X-ray passes through the centre of the

pixel[8].

Fourier Slice Theorem claims that the 1D Fourier

transform of the projections is equal to the 2D Fourier

transform of the image evaluated on the line that the

projection was taken on. In this paper, it is formulated that

the 2D Fourier values of the scanning object can be obtained

exactly through Mojette transform. Let (𝑢, 𝑣) be the

coordinate in the frequency domain, then the 2D discrete

Fourier transform of 𝑓(𝑥, 𝑦) is:

𝐹 𝑢, 𝑣 = 𝑓(𝑥, 𝑦)𝑒−𝑗2𝜋

𝑁(𝑢𝑥 +𝑣𝑦)

𝑦𝑥 (3)

The 1D discrete Fourier transform of 𝑀p,q(𝜌) is:

𝐹𝑀𝑝 ,𝑞 𝑟 = 𝑀𝑝 ,𝑞(𝜌)𝜌 𝑒−𝑗2𝜋

𝐿𝜌𝑟

(4)

where 𝐿 is the projection size, and 𝐿 = 𝑁 − 1 𝑝 +𝑞+1. Combining (2) and (4), we can get:

𝐹𝑀𝑝 ,𝑞 𝑟 = 𝑓(𝑥, 𝑦)𝑒−𝑗2𝜋

𝐿𝑟(𝑝𝑥 +𝑞𝑦 )

𝑦𝑥 (5)

Comparing (3) and (5), it is hard to map 𝐹𝑀𝑝 ,𝑞 𝑟 to

𝐹 𝑢, 𝑣 directly. Considering the periodicity of Fourier

transform, we can merge the items gained by (2) with the

same 𝑚𝑜𝑑(𝑝𝑥 + 𝑞𝑦, 𝑁), then projection size can be reduced

to 𝑁. So

𝑀𝑝 ,𝑞′ 𝜌 = 𝑓(𝑥, 𝑦)|𝜌=𝑚𝑜𝑑 (𝑝𝑥 +𝑞𝑦 ,𝑁)𝑦𝑥 (6)

Its 1D Fourier transform is:

𝐹𝑀𝑝 ,𝑞′ = 𝑓 𝑥, 𝑦 𝑒−𝑗

2𝜋

𝑁𝑟 𝑚𝑜𝑑 𝑝𝑥 +𝑞𝑦 ,𝑁

𝑦𝑥

= 𝑓(𝑥, 𝑦)𝑒−𝑗2𝜋

𝑁[𝑚𝑜𝑑 𝑟𝑝𝑥 +𝑟𝑞𝑦 ,𝑁 ]

𝑦𝑥

(7)

The mapping relationship is revealed over the analysis of

(3) and (7):

∀𝑥, 𝑦, 𝑚𝑜𝑑 𝑢𝑥 + 𝑣𝑦 − 𝑟𝑝𝑥 − 𝑟𝑞𝑦, 𝑁 = 0 (8)

Hence, the 1D Fourier transform of the modified Mojette

projections taken over angles 𝜃 = 𝑡𝑎𝑛−1(𝑞 𝑝 ) can be

mapped to the 2D Fourier plane of the scanning object with

the mapping relationship:

𝑚𝑜𝑑 𝑢 − 𝑟𝑝 = 0 & 𝑚𝑜𝑑 𝑣 − 𝑟𝑞, 𝑁 = 0 (9)

As a result, we can get the partial Fourier coefficients

where 𝑚𝑜𝑑 𝑞𝑢 − 𝑝𝑣, 𝑁 = 0 . The Fourier coefficients

gained through the mapping of the Mojette projections in

Fig. 1. (a) (𝑁 = 6, 𝑡𝑎𝑛𝜃 = 1/2) are marked with star in Fig.

1(b).

(a) Dirac-Mojette Projection of a 6×6 image

(b) The corresponding Fourier values of the Mojette projections

Fig. 1. The geometry of Mojette transform and the mapping relationship in

Fourier domain

B. Compressed Sensing

Compressed sensing is a recently developed theory of

signal recovery from highly incomplete information, and

thus will be employed here to reconstruct the image from

the partial Fourier coefficients. The central idea of CS is that

a sparse or compressible signal 𝑥 ∈ 𝘙𝑁 can be recovered

from a small number of linear measurements 𝑏 = 𝐴𝑥 ∈ 𝘙𝐾 , 𝐾 ≪ 𝑁. The reconstruction can be achieved by solving

the well-known basis pursuit problem [9]-[10]:

min ||𝑥||1 𝑠. 𝑡. 𝐴𝑥 = 𝑏

With the noisy and incomplete samples, an appropriate

relaxation is given by:

min ||𝑥||1 𝑠. 𝑡. | 𝑏 − 𝐴𝑥 |2 ≤ 𝜍

where 𝜍 > 0 is related to the noise.

To gain more effective reconstruction results, a simple

and fast algorithm called RecPF (Reconstruction from

Partial Fourier data) [11] is adopted. It uses an alternating

minimization scheme to solve the following model:

min𝐼 𝑇𝑉 𝐼 + 𝜆||𝜓𝐼||1 + 𝜇||𝐹𝑝 𝐼 − 𝑓𝑝 ||2

where 𝐼 is the image to be reconstructed, 𝑇𝑉(𝐼) is the total

variation regularization term, 𝜓 is a sparsifying basis (e.g.,

wavelet basis), 𝐹𝑝 is a partial Fourier matrix and 𝑓𝑝 denotes

the partial Fourier coefficients. As the main computation of

solving the model only involves shrinkage and fast Fourier

transforms, the reconstruction process is quite fast.

84


C. Proposed Frame

The proposed frame contains two stages. At the first

stage, the sinogram is converted to Mojette projections using

the linear system designed in paper [3], then their Fourier

values are mapped to the 2D Fourier domain of the object.

At the second stage, CS is applied to the partial Fourier

coefficients for the recovery of the object. The flowchart is

shown as below.

Fig. 2. Flowchart of the proposed method

Based on the above analysis, it is known that the exact

Fourier coefficients can be obtained through Mojette

transform, which is very beneficial for the reconstruction

accuracy. The application of CS afterwards reduces the

Fourier samples needed for a perfect recovery. Hence, good

reconstruction results can be expected using the proposed

frame when the CT scanning just covers limited number of

view angles.

III. EXPERIMENTAL RESULTS AND DISCUSSION

A. Noise-free reconstruction

In this part, two groups of experiments are designed over

the same Farey angle sets and different Farey angle sets,

respectively. The performance of different methods is

measured through the quantities listed below: the error in the

reconstructed image relative to the original image (Err),

mean square error (MSE) and signal-to-noise ratio (SNR).

Let 𝐼0 denote the original image, 𝐼 the reconstructed image,

then the definitions of the two quantities are:

𝐸𝑟𝑟 = [𝐼 𝑖, 𝑗 − 𝐼0(𝑖, 𝑗)]𝑗𝑖

𝐼0(𝑖, 𝑗)2𝑗𝑖

𝑀𝑆𝐸 = 𝑚𝑒𝑎𝑛[(𝐼 𝑖, 𝑗 − 𝐼0 𝑖, 𝑗 )2]

𝑆𝑁𝑅 = 10 × 𝑙𝑔 (𝐼 𝑖, 𝑗 − 𝑚𝑒𝑎𝑛(𝐼))2

𝑗𝑖

(𝐼 𝑖, 𝑗 − 𝐼0(𝑖, 𝑗))2𝑗𝑖

Firstly, the experiments are conducted over the same

Farey angle sets. We present simulation results of the

256 × 256 phantom image reconstrucion obtained by MCS

and two other methods: FBP and Mift, where FBP uses

filtered backprojection algorithm over the Radon projections

and Mift applies direct inverse Fourier transform to the

Mojette projections. The projections are taken over the

Farey angles. Fig.3 shows the reconstruction results of the

three methods for 𝐹8 projetions, from which we can see that

FBP result has disturbing artifacts due to limited projections,

and Mift can't recover the image from insufficient Fourier

samples. It is obvious that MCS does the almost exact

construction. The Err of the three different methods as a

function of the order of Farey series is plot in Fig.4. We can

see that MCS has already got the good reconstruction result

with 𝐹4 projections, where the Err of Mift and FBP is still

larger than 50%. Fig.4 also shows that FBP can gain better

results than Mift with the increasing number of projections.

(a) original phantom (b) FBP

(c) Mift (d) MCS

Fig. 3. Reconstruction results of the three methods from the projections

taken over F8

Fig. 4. A comparison of Err gained by different methods with the increasing

order of the Farey series

Then the experiments are conducted over different Farey

angle sets. The Mojette filtered backprojection algorithm

(MFBP) [12] is used here for comparison. We use the same

experimental image as MFBP, a 128 × 128 phantom image

consisting of a 17 × 17 square object with unitary value

whereas boundaries are only half valued. Table I lists the

MSE of MCS and MFBP reconstruction results with

different orders of Farey series. We can see that MCS can

gain better results over 𝐹1 angle set with just 4 projection

lines than MFBP over 𝐹32 angle set with 1296 projection

lines. Here are the reconstruction results in Fig.5. It is shown

that the result of MFBP over 𝐹32 angle in (b) has some

disturbing artefacts, while MCS gives clear and exact

85


reconstruction result with only 4 projection lines. Fig.5 (d)

shows the line mask of the partial Fourier coefficients which

are the input of CS. If the X-ray tube current is fixed, the

total radiation exposure of the target is proportional to the

number of view angles. Hence, the conclusion is drawn that

less projections are needed for the reconstruction with the

help of CS, which means shorter scanning time and lower

radiation dosage, and thus will benefit the patients more.

(a) original phantom (b) MFBP over 𝐹32 (c) MCS over F1 (d) line mask

Fig. 5. Reconstruction results of MCS and MFBP

TABLE I: MSE OF MCS AND MFBP RECONSTRUCTION RESULTS

Method MCS MFBP

F 𝐹1 𝐹2 𝐹3 𝐹4 𝐹32 𝐹64 𝐹128

#proj 4 8 16 24 1296 5040 20088

MSE 1.69×10-7 6.95×10-8 7.88×10-9 1.23×10-10 0.01322 2×10-5 0

B. Noise tolerance

This experiment is designed to test the noise tolerance of

the proposed frame. We generate our test sets using the

256×256 Shepp-Logan phantom image and FBP is

employed here for comparison. The Gaussian noise is added

to the Mojette projections in MCS and the Radon

projections in FBP, respectively. In both methods, the

projections are taken over F8 angle set, i.e., 88 projections.

As the noise is produced randomly, the quantitative

assessment (Err and SNR) is the mean value of the 20

groups of experimental data, and the results are shown in

Table II. We can see that for the Gaussian noise (0,0.001),

i.e. with mean 0 and standard deviation 0.001, MCS can

gain a much better reconstruction than FBP, the results of

which are shown in Fig.6. But for (0,0.01) noise, the result

of FBP barely changes while the reconstruction quality of

MCS reduces a lot, which means FBP is robust to noise and

MCS is sensitive. The reason is that MCS is based on noise-

sensitive Fourier transform. When (0, 0.01) noise is added to

the projections, the noise of its Fourier coefficients becomes

(0,0.1) for a 256×256 image, verified in Fig.7. Since CS has

certain ability to suppress noise, we get the conclusion that

though sensitive, the proposed frame can deal with small

noise effectively. It will be the future work to deal with the

noisier projections [13].

TABLE II: NOISE RESPONSE OF FBP AND MCS

Noise FBP MCS

Err SNR Err SNR

(0,0.001) 37.1577% 7.3586 2.1929% 31.9393

(0,0.01) 37.1587% 7.3584 23.1656% 11.4631

(a) FBP (b) MCS

Fig. 6. Reconstruction results from the F8 projections with Gaussian noise

(0,0.001)

(a) the noise of projections

(b) the corresponding noise of the Fourier values

Fig. 7. Noise of the projections and the Fourier values

86


IV. CONCLUSION

In this paper, a novel frame for parallel-beam CT

reconstruction is presented. Firstly, the sinogram is

converted to the projections gained through Mojette

transform, an exact discretization of Radon transform. On

each view angle, the projections are summed up under some

principles. Then the 1D Fourier coefficients of the merged

projections are mapped to the 2D Fourier domain of the

object. Finally compressed sensing is employed to deal with

the partial Fourier coefficients and can recover the object

very well and suppress the small noise effectively.

Experimental results have demonstrated the advantages of

the proposed method. With the presence of Mojette

transform and compressed sensing, the purpose of reducing

the radiation dosage during CT examinations without

compromising the image quality is achieved.

REFERENCES

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[4] H. Fayad, J. P. Guedon, I. Svalbe, Y. Bizais, and N. Normand,

“Applying Mojette discrete Radon transforms to classical

tomographic data,” in Proc. of SPIE, 2008, vol. 6913, pp. 1605-7422.

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from projections,” Lect. Notes Biomath, vol. 26, Springer-Verlag,

Berlin, Heidelberg, New York, 1979.

[6] M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed

sensing MRI,” IEEE Signal Processing Magazine, pp. 72-82, 2008.

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[8] S. Chandra, N. Normanda, A. Kingston, J. Guedon, and I. Svalbe.

(2010). Fast Mojette transform for discrete tomography. Elsevier

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[9] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles:

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information,” IEEE Trans. Inform. Theory, vol. 52, no. 2, pp. 489-

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[10] D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of

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[12] M. Servieres, N. Normand, J. P. Guedon, and Y. Bizais, “The Mojette

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[13] B. Recur, P. Desbarats, and J. P. Domenger, “Mojette reconstruction

from noisy projections,” in Proc. 2010 2nd International Conference

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201-206.

Wen Hou received the B.Eng. degree from Nanjing

University of Aeronautics and Astronautics, China.

She is currently doing Ph.D. in the Faculty of

Engineering and Industrial Sciences, Swinburne

University of Technology, Australia. Her research

interests include medical imaging and

reconstruction.

Cishen Zhang received the B.Eng. degree from

Tsinghua University, China, in 1982 and Ph.D. degree

in Electrical Engineering from Newcastle University,

Australia, in 1990. Between 1971 and 1978, he was an

Electrician with Changxindian (February Seven)

Locomotive Manufactory, Beijing, China. He carried

out research work on control systems at Delft

University of Technology, The Netherlands, from

1983 to 1985. After his Ph.D. study from 1986 to 1989 at Newcastle

University, he was with the Department of Electrical and Electronic

Engineering at the University of Melbourne, Australia as a Lecturer, Senior

Lecturer and Associate Professor and Reader till October 2002. He is

currently with the Faculty of Engineering and Industrial Sciences,

Swinburne University of Technology, Australia. His research interests

include signal processing, medical imaging and control.

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